Random Attractors of a Stochastic Hopfield Neural Network Model with Delays

Abstract

The global asymptotic behavior of a stochastic Hopfield neural network model (HNNM) with delays is explored by studying the existence and structure of random attractors. It is firstly proved that the trajectory field of the stochastic delayed HNNM admits an almost sure continuous version, which is compact for \(t>\tau \) (where \(\tau \) is the delay) by a construction based on the random semiflow generated by the diffusion term due to Mohammed ( Stoch. Stoch. Rep. 29: 89–131, 1990). Then, this version is shown to generate a random dynamical system (RDS) by a Wong–Zakai approximation, after which the existence of a random absorbing set is obtained via uniform apriori estimate of the solutions. Subsequently, the pullback asymptotic compactness of the RDS generated by the stochastic delayed HNNM is established and hence the existence of random attractors is obtained. Sufficient conditions under which the attractors turn out to be an exponential attracting stationary solution are also given. Finally, some numerical simulations illustrate the results.